| L(s) = 1 | − i·3-s + 3.09·5-s − 0.646i·7-s − 9-s + 1.41·11-s + (2.44 − 2.64i)13-s − 3.09i·15-s + 3.58·17-s + 1.11·19-s − 0.646·21-s − 3.46·23-s + 4.58·25-s + i·27-s + 0.913i·29-s + 1.93i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.38·5-s − 0.244i·7-s − 0.333·9-s + 0.426·11-s + (0.679 − 0.733i)13-s − 0.799i·15-s + 0.868·17-s + 0.256·19-s − 0.140·21-s − 0.722·23-s + 0.916·25-s + 0.192i·27-s + 0.169i·29-s + 0.348i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.556167851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.556167851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 13 | \( 1 + (-2.44 + 2.64i)T \) |
| good | 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 0.646iT - 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 0.913iT - 29T^{2} \) |
| 31 | \( 1 - 1.93iT - 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5.95iT - 41T^{2} \) |
| 43 | \( 1 + 3.58iT - 43T^{2} \) |
| 47 | \( 1 + 3.74iT - 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 0.913iT - 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 2.44iT - 71T^{2} \) |
| 73 | \( 1 + 5.65iT - 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 8.78iT - 89T^{2} \) |
| 97 | \( 1 - 7.89iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.922249663671908249080148767216, −7.943850666896921111496199501830, −7.29609690249995731734203307896, −6.24715924112770650671111852714, −5.89866225020810202887914136931, −5.12614946603704832365374010357, −3.83888480967276077770262712648, −2.86279232699078608607904724324, −1.83063532270804232912146395668, −0.964381111754874319135321554340,
1.30821403230402996784600395630, 2.26996944205890745248460943220, 3.32945762899998192981705078298, 4.29722101586598828480843410206, 5.19507126483204635657925579925, 6.09890997105474956728329407306, 6.29269971297072976234435386727, 7.58128891567439956941954886535, 8.464432724355293856944834278082, 9.334683474069541935942025870207
